Fractal Dimension of Riemann–liouville Fractional Integral of Certain Unbounded Variational Continuous Function

Li, Yang; Xiao, Wei

doi:10.1142/s0218348x17500475

Cited by 16 publications

(6 citation statements)

References 11 publications

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“…There are many works to construct a curve of 1-dimensional fractal continuous function, then to calculate its fractional integral dimension [2] [14] [16] [17] [24]…”

Section: On All 1-dimensional Fractal Continuous Functionsmentioning

confidence: 99%

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A Technique for Estimation of Box Dimension about Fractional Integral

Zhang

2023

APM

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“…There are many works to construct a curve of 1-dimensional fractal continuous function, then to calculate its fractional integral dimension [2] [14] [16] [17] [24]…”

Section: On All 1-dimensional Fractal Continuous Functionsmentioning

confidence: 99%

“…(See [2] [19]). Liang [3] and Liu [20] investigated the relationship of 1-dimensional continuous function ( ) f x with its Riemann-Liouville integral, and proved that ( )…”

Section: Introductionmentioning

confidence: 99%

A Technique for Estimation of Box Dimension about Fractional Integral

Zhang

2023

APM

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“…Barnsley and Ruan have made research on the linear fractal interpolation functions in [5,6], respectively. Moreover, there exist certain particular examples of one-dimensional fractal functions discussed in [7][8][9][10][11][12][13][14] and two-dimensional fractal functions constructed in [15,16]. For Hölder continuous functions, ref.…”

Section: Introductionmentioning

confidence: 99%

On the Lower and Upper Box Dimensions of the Sum of Two Fractal Functions

Liang

2022

Fractal Fract

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Let f and g be two continuous functions. In the present paper, we put forward a method to calculate the lower and upper Box dimensions of the graph of f+g by classifying all the subsequences tending to zero into different sets. Using this method, we explore the lower and upper Box dimensions of the graph of f+g when the Box dimension of the graph of g is between the lower and upper Box dimensions of the graph of f. In this case, we prove that the upper Box dimension of the graph of f+g is just equal to the upper Box dimension of the graph of f. We also prove that the lower Box dimension of the graph of f+g could be an arbitrary number belonging to a certain interval. In addition, some other cases when the Box dimension of the graph of g is equal to the lower or upper Box dimensions of the graph of f have also been studied.

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“…(2) Particular functions like Weierstrass and their variant types (some called Besicovitch or Besicovitch type functions) such as [1,3,8,17,18,23]. (3) Fractal integral or derivative on these particular functions such as [7,9,13]. Here it's worthy to mention that Box dimension of Weierstrass function was calculated much early (see Example 3.4).…”

Section: Introductionmentioning

confidence: 99%

On Fractal Features and Fractal Linear Space About Fractal Continuous Functions

Xiao¹

2021

Preprint

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This paper investigates fractal dimension of linear combination of fractal continuous functions with the same or different fractal dimensions. It has been proved that: (1) BV I all fractal continuous functions with bounded variation is fractal linear space; (2) 1 D I all fractal continuous functions with Box dimension one is a fractal linear space; (3) s D I all fractal continuous functions with identical Box dimension s(1 < s ≤ 2) is surprisingly a non-fractal linear space, even non-fractal linear manifold, beyond our initial expectation, because the Box dimension of linear combination of fractal continuous functions can take any real number in [1, s) if it exists, and some different upper and lower Box dimension if it does not exit. This attracts our interests to probe into fractal characteristics of s D I , and get some suggesting results.

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Fractal Dimension of Riemann–liouville Fractional Integral of Certain Unbounded Variational Continuous Function

Cited by 16 publications

References 11 publications

A Technique for Estimation of Box Dimension about Fractional Integral

A Technique for Estimation of Box Dimension about Fractional Integral

On the Lower and Upper Box Dimensions of the Sum of Two Fractal Functions

On Fractal Features and Fractal Linear Space About Fractal Continuous Functions

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